Determining simplicity and computing topological change in strongly normal partial tilings of R2 or R3

A convex polygon in R-2, or a convex polyhedron in R-3, Will be called a tile. A connected set P of tiles is called a partial tiling if the intersection of any two of the tiles is either empty, or is a vertex or edge (in R-3: or face) of both. P is called strongly normal (SN) if, for any partial tiling P subset of or equal to P and any tile P is an element of P, the neighborhood N(P, P) of P (the union of the tiles of P' that intersect P) is simply connected. Let P be SN, and let N*(P, P) be the excluded neighborhood of P in P (i.e., the union of the tiles of P, other than P itself, that intersect P). We call P simple in P if N(P, P) and N*(P, P) are topologically equivalent. This paper presents methods of determining, for an SN partial tiling P, whether a tile P is an element of P' is simple, and if not, of counting the numbers of components and holes tin R3: components, tunnels and cavities) in N*(P, P). (C) 1999 Pattern Recognition Society. Published by Elsevier Science Ltd. All rights reserved.